Source Parameters from Identified Hadron Spectra and HBT Radii for AuAu Collisions at sNN 200 GeV in

1
Source Parameters from Identified Hadron Spectra
and HBT Radii for Au-Au Collisions at ?sNN 200
GeV in PHENIX

• J.M. Burward-Hoy
• Lawrence Livermore National Laboratory
• for the
• PHENIX Collaboration

Motivation

• The objective is to measure the characteristics
  of the particle emitting source from both spectra
  and HBT radii simultaneously.
• An interpretation of the data assuming
  relativistic hydrodynamic expansion is presented.
• The study uses PHENIX Preliminary data as
  presented by T. Chujo and A. Enokizono.

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Single ??K?p? SpectraPHENIX Time of Flight
Detecting ??, K?, p? in PHENIX
HBT AnalysisPHENIX Electromagnetic Calorimeter
momentum resolution ?p/p 1 ? 1 p
TOF resolution 120 ps
EMCal resolution 450 ps
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A Simple Model for the Source

• Model by Wiedemann, Scotto, and Heinz , Phys.
  Rev. C 53, 918 (1996)
• E. Schnedermann, J. Sollfrank, and U. Heinz,
  Phys. Rev. C 48, 2462 (1993)
• Fluid elements each in local thermal equilibrium
  move in space-time with hydrodynamic expansion.
• No temperature gradients
• Boost invariance along collision axis z.
• Infinite extent along rapidity y.
• Cylindrical symmetry with radius r.
• Particle emission
• hyperbola of constant proper time ?0
• Short emission duration
• ?t lt 1 fm/c

r
t
z
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Transverse Kinetic Energy Spectra
5 Au-Au at ?sNN 200 GeV

• Kinetic energy spectra broaden with increasing
  mass, from ? to K to p (they are not parallel).

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Mean Transverse Momentum
Npart

• Mean pt increases with Npart and m0, indicative
  of radial expansion.
• Relative increase from peripheral to central
  greater for (anti)p than for ?, K.
• Systematic uncertainties ? 10, K 15, and
  (anti-)p 14

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Reproducing the Shape of the Single Particle
Spectra
Radial position on freeze-out surface ?
r/R Particle density distribution f(?) is
independent of ?
Shape of spectra important
?t

f(?)
?
parameters normalization A freeze-out
temperature Tfo surface velocity ?T
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Linear flow profile ?(?) ?T? lt?T gt 2?T/3 S.
Esumi, S. Chapman, H. van Hecke, and N. Xu,
Phys. Rev. C 55, R2163 (1997)
Minimize contributions from hard processes
(mt-m0) lt 1 GeV Exclude ? resonance region
pT lt 0.5 GeV/c
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Fitting the Transverse Momentum Spectra

• Simultaneous fit in range (mt -m0 ) lt 1 GeV is
  shown.
• The top 5 centralities are scaled for visual
  clarity.
• Similar fits for negative particles.

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Fitting the Transverse Momentum Spectra

• Simultaneous fit in range (mt -m0 ) lt 1 GeV is
  shown.
• The top 5 centralities are scaled for visual
  clarity.
• Similar fits for positive particles.

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For All Centralities?2 Contours in Parameter
Space Tfo and ?T
PHENIX Preliminary
In each centrality, the first 20 n-? contour
levels are shown. From the most peripheral to the
most central data, the single particle spectra
are fit simultaneously for all pions, kaons, and
protons.
PHENIX Preliminary
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A close-up Most Central and Most Peripheral
For the 5 spectra ?T 0.7 0.2 syst. Tfo
(110 ? 23 syst.) MeV
For the most peripheral spectra ?T 0.46 0.02
stat. ?0.2 syst. Tfo 135 ? 3 stat. ? 23
syst. MeV
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The Parameters Tfo and ?T vs. Npart

• Expansion parameters in each centrality
• Overall systematic uncertainty is shown.
• A trend with increasing Npart is observed
• Tfo and ?T
• Saturates at mid-central

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Expansion from the kT Dependence of HBT Radii
Use analytical forms for the radii from
Wiedemann, Scotto, and Heinz
Ref PRC 53 (No. 2), Feb. 1996
T 150 MeV, R 3 fm, ?0 3 fm/c
RL (fm)
Ro (fm)
Rs (fm)
PHENIX Preliminary
??
Ro (fm)
RL (fm)
parameters geometric radius R freeze-out
temperature Tfo flow rapidity at surface
?T freeze-out proper time ?0
Rs (fm)
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Fitting the kT Dependence of HBT Radii

• ?2 contours in parameter space Tfo and ?T
• The contours are not closed
• In this region of parameter space, the minimum ?2
  value is found
• These contours are the n-? values relative to
  this minimum

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HBT Radii and Single Particle Spectra
PHENIX Preliminary

• Spectra and RL are consistent within 2.5 ?.
• Ro and Rs disagree with the spectra.
• Rs prefers high flow and low temperatures
• ?T gt 1.0 and Tfo lt 50 MeV
• Ro prefers
• ?T gt 1.4 and Tfo gt 100 MeV
• (R-contours not closed)

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Using spectra information to constrain HBT fits
From the spectra (systematic errors) ?T 0.7
0.2 syst. Tfo 110 ? 23 syst. MeV
PHENIX Preliminary
Rs (fm)
Ro (fm)
RL (fm)
??
?0 13?2 fm/c
R 9.60.2 fm

• 10 central positive pion HBT radii (similar
  result for negative pion data).
• Systematic uncertainty in the data is 8.2 for
  Rs, 16.1 for Ro, 8.3 for RL.

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Using spectra information to constrain HBT fits
From the spectra (systematic errors) ?T 0.7
0.2 syst. Tfo 110 ? 23 syst. MeV
PHENIX Preliminary
Rs (fm)
Ro (fm)
RL (fm)
?-?-
R 9.70.2 fm
?0 13?2 fm/c

• 10 central negative pion HBT radii.
• Systematic uncertainty in the data is 8.2 for
  Rs, 16.1 for Ro, 8.3 for RL.

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Conclusions

• Expansion measured from spectra depends on Npart.
• Saturates to constant for most central.
• Used simple profiles for the expansion and
  particle density distribution
• Linear velocity profile
• Flat particle density
• Within this hydro model, no common source
  parameters could be found for spectra and all HBT
  radii simultaneously.

For the most peripheral spectra ?T 0.5 ?0.2
syst. (lt ?Tgt 0.3 0.2 syst. ) Tfo 135 ? 23
MeV
to the 5 spectra ?T 0.7 0.2 syst. (lt
?Tgt 0.5 0.2 syst.) Tfo 110 ? 23 MeV
Rs prefers ?T gt 1.0 and Tfo lt 50 MeV Ro prefers
?T gt 1.4 and Tfo gt 100 MeV
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Particle Yields per Npart pair vs. Npart
PHENIX Preliminary

• Yield per pair Npart shown on a log scale for
  visual clarity only.
• Linear dependence on Npart.
• Relative increase from peripheral to central
  greater for K than for ?, (anti-)p.
• Systematic uncertainties shown as lines.

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Year-1 Mean transverse momentum
20/- 5 increase
20/- 5 increase
Open symbols pp collisions

• Mean pt ? with Npart , m0 ? radial flow
• Relative increase from peripheral to central same
  for ?, K, (anti)p
• (Anti)proton significant ? from pp collisions

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Year-1Fitting the Single Particle Spectra
130 GeV
Simultaneous fit (mt -m0 ) lt 1 GeV (see arrows)
PHENIX Preliminary
PHENIX Preliminary
Exclude ? resonances by fitting pt gt 0.5
GeV/c The resonance region decreases T by 20
MeV. This is no surprise! Sollfrank and Heinz
also observed this in their study of SS
collisions at CERN energies. NA44 also had a
lower pt cut-off for pions in PbPb collisions.
PHENIX Preliminary
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Year-1 Single Particle Spectra
130 GeV
PHENIX Preliminary
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Year-1 An example of a fit to ??. . .
PHENIX Preliminary
130 GeV
The ?2 is better for the higher order fit. . .
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HBT Radii and Single Particle Spectra

• HBT radii suggest a lower temperature and higher
  flow velocity
• Use best fit of singles and convert ? to ?
• Singles and HBT radii are within 2?

130 GeV
Tfo
PHENIX Preliminary
?f
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From HydrodynamicsRadial Flow Velocity Profiles
Velocity profile at ? 20 fm/c
freeze-out hypersurface
At each snapshot in time during the expansion,
there is a distribution of velocities that vary
with the radial position r
Plot courtesy of P. Kolb
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Hydrodynamics-based parameterization
1/mt dN/dmt A ? f(?) ? d? mT K1( mT /Tfo cosh
? ) I0( pT /Tfo sinh ? )
?t
integration variable ? ? radius r r/R
definite integral from 0 to 1 particle density
distribution f(?) const

• parameters
• normalization A
• freeze-out temperature Tfo
• surface velocity ?t

linear velocity profile ?t(?) ?t? surf.
velocity ?t ave. velocity lt?t gt 2/3 ?t
boost ?(?) atanh( ?t(?) )
minimize contributions from hard processes fit
mt-m0lt1 GeV
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Two Approaches to Calculating HBT Radii. . .

• (After assuming something about the source
  function. . .)
• A numerical approach is to
• numerically determine C(K,q) from S(K,q)
• C(K,q) 1 exp -qs2Rs2(K) q02Ro2(K)
  ql2Rl2(K)-2qlqoRlo2(K)
• there is an exact calculation of these radii
  (full integrations)
• there are lower-order and higher-order
  approximations (from series expansion of Bessel
  functions).
• The lowest-order form for Rs was used in Phenix
  PRL. (A similar expression is used by NA49).
• The higher-order approximation is very good when
  compared to the exact calculation for Rs and RL.

What Im doing
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Important Assumptions Used. . .

• Integration over ? is done exactly.
• Boost invariance (vL z/t). Space-time rapidity
  equals flow rapidity
• Infinitely long in y.
• In LCMS, y and ?L 0.
• Integrals expressed in terms of the modified
  Bessel functions
• For HBT radii, approximations are used in
  integration over x and y.
• Saddle point integration using approximate
  saddle point
• Series expansion of Bessel functions
• Assume mT/Tgt1

As is also assumed in calculating particle
spectra
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Calculating the HBT Radii
Linear flow rapidity profile Defined weight
function Fn
?f 0
Rs
?f 0.3
?f 0.6
?f 0.9
T 150 MeV, R 3 fm, ?0 3 fm/c
parameters geometric radius R freeze-out
temperature T flow rapidity at surface
?f freeze-out proper time ?0
Constants are determined up to order 3 from
Bessel function expansion
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The Analytical Evaluation of the HBT Radii
Linear flow rapidity profile
Up to order 3 in Bessel function expansion